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Transverse waves: Introduction
∗
and key concepts (Grade 10) [NCS]
Free High School Science Texts Project
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0
†
1 Introduction
Waves occur frequently in nature. The most obvious examples are waves in water, on a dam, in the ocean, or
in a bucket. We are most interested in the properties that waves have. All waves have the same properties,
so if we study waves in water, then we can transfer our knowledge to predict how other examples of waves
will behave.
2 What is a transverse wave ?
We have studied pulses in Transverse Pulses, and know that a pulse is a single disturbance that travels
through a medium. A wave is a periodic, continuous disturbance that consists of a train of pulses.
Denition 1: Wave
A wave is a periodic, continuous disturbance that consists of a train of pulses.
Denition 2: Transverse wave
A transverse wave is a wave where the movement of the particles of the medium is perpendicular
to the direction of propagation of the wave.
2.1 Investigation : Transverse Waves
Take a rope or slinky spring. Have two people hold the rope or spring stretched out horizontally. Flick the
one end of the rope up and down continuously to create a train of pulses.
Figure 1
1. Describe what happens to the rope.
∗ Version
1.1: Aug 5, 2011 10:21 am -0500
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2. Draw a diagram of what the rope looks like while the pulses travel along it.
3. In which direction do the pulses travel?
4. Tie a ribbon to the middle of the rope. This indicates a particle in the rope.
Figure 2
5. Flick the rope continuously. Watch the ribbon carefully as the pulses travel through the rope. What
happens to the ribbon?
6. Draw a picture to show the motion of the ribbon. Draw the ribbon as a dot and use arrows to indicate
how it moves.
In the Activity, you have created waves. The medium through which these waves propagated was the rope,
which is obviously made up of a very large number of particles (atoms). From the activity, you would have
noticed that the wave travelled from left to right, but the particles (the ribbon) moved only up and down.
Figure 3:
A transverse wave, showing the direction of motion of the wave perpendicular to the direction
in which the particles move.
When the particles of a medium move at right angles to the direction of propagation of a wave, the wave
is called transverse. For waves, there is no net displacement of the particles (they return to their equilibrium
position), but there is a net displacement of the wave. There are thus two dierent motions: the motion of
the particles of the medium and the motion of the wave.
The following simulation will help you understand more about waves. Select the oscillate option and
then observe what happens.
Phet simulation for Transverse Waves
This media object is a Flash object. Please view or download it at
<http://cnx.org/content/m40097/1.1/wave-on-a-string.swf>
Figure 4
2.2 Peaks and Troughs
Waves have moving peaks (or crests ) and troughs. A peak is the highest point the medium rises to and a
trough is the lowest point the medium sinks to.
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Peaks and troughs on a transverse wave are shown in Figure 5.
Figure 5:
Peaks and troughs in a transverse wave.
Denition 3: Peaks and troughs
A peak is a point on the wave where the displacement of the medium is at a maximum. A point
on the wave is a trough if the displacement of the medium at that point is at a minimum.
2.3 Amplitude and Wavelength
There are a few properties that we saw with pulses that also apply to waves. These are amplitude and
wavelength (we called this pulse length).
Denition 4: Amplitude
The amplitude is the maximum displacement of a particle from its equilibrium position.
2.3.1 Investigation : Amplitude
Figure 6
Fill in the table below by measuring the distance between the equilibrium and each peak and troughs in the
wave above. Use your ruler to measure the distances.
Peak/Trough
a
b
c
d
e
f
Measurement (cm)
Table 1
1. What can you say about your results?
2. Are the distances between the equilibrium position and each peak equal?
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3. Are the distances between the equilibrium position and each trough equal?
4. Is the distance between the equilibrium position and peak equal to the distance between equilibrium
and trough?
As we have seen in the activity on amplitude, the distance between the peak and the equilibrium position
is equal to the distance between the trough and the equilibrium position. This distance is known as the
amplitude of the wave, and is the characteristic height of wave, above or below the equilibrium position.
Normally the symbol A is used to represent the amplitude of a wave. The SI unit of amplitude is the metre
(m).
Figure 7
Exercise 1: Amplitude of Sea Waves
(Solution on p. 10.)
If the peak of a wave measures 2 m above the still water mark in the harbour, what is the amplitude
of the wave?
2.3.2 Investigation : Wavelength
Figure 8
Fill in the table below by measuring the distance between peaks and troughs in the wave above.
Distance(cm)
a
b
c
d
Table 2
1.
2.
3.
4.
What can you say about your results?
Are the distances between peaks equal?
Are the distances between troughs equal?
Is the distance between peaks equal to the distance between troughs?
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As we have seen in the activity on wavelength, the distance between two adjacent peaks is the same no
matter which two adjacent peaks you choose. There is a xed distance between the peaks. Similarly, we
have seen that there is a xed distance between the troughs, no matter which two troughs you look at. More
importantly, the distance between two adjacent peaks is the same as the distance between two adjacent
troughs. This distance is called the wavelength of the wave.
The symbol for the wavelength is λ (the Greek letter lambda) and wavelength is measured in metres (m).
Figure 9
Exercise 2: Wavelength
(Solution on p. 10.)
The total distance between 4 consecutive peaks of a transverse wave is 6 m. What is the wavelength
of the wave?
2.4 Points in Phase
2.4.1 Investigation : Points in Phase
Fill in the table by measuring the distance between the indicated points.
Figure 10
Points Distance (cm)
A to F
B to G
C to H
D to I
E to J
Table 3
What do you nd?
In the activity the distance between the indicated points was the same. These points are then said to be
in phase. Two points in phase are separate by an integer (0,1,2,3,...) number of complete wave cycles. They
do not have to be peaks or troughs, but they must be separated by a complete number of wavelengths.
We then have an alternate denition of the wavelength as the distance between any two adjacent points
which are in phase.
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Denition 5: Wavelength of wave
The wavelength of a wave is the distance between any two adjacent points that are in phase.
Figure 11
Points that are not in phase, those that are not separated by a complete number of wavelengths, are
called out of phase. Examples of points like these would be A and C , or D and E , or B and H in the
Activity.
2.5 Period and Frequency
Imagine you are sitting next to a pond and you watch the waves going past you. First one peak arrives, then
a trough, and then another peak. Suppose you measure the time taken between one peak arriving and then
the next. This time will be the same for any two successive peaks passing you. We call this time the period,
and it is a characteristic of the wave.
The symbol T is used to represent the period. The period is measured in seconds (s).
Denition 6: The period (T) is the time taken for two successive peaks (or troughs)
to pass a xed point.
Imagine the pond again. Just as a peak passes you, you start your stopwatch and count each peak going
past. After 1 second you stop the clock and stop counting. The number of peaks that you have counted in
the 1 second is the frequency of the wave.
Denition 7: The frequency is the number of successive peaks (or troughs) passing a
given point in 1 second.
The frequency and the period are related to each other. As the period is the time taken for 1 peak to
pass, then the number of peaks passing the point in 1 second is T1 . But this is the frequency. So
f=
1
T
(1)
T =
1
.
f
(2)
or alternatively,
For example, if the time between two consecutive peaks passing a xed point is 12 s, then the period of the
wave is 12 s. Therefore, the frequency of the wave is:
f
=
=
=
1
T
1
1
2 s
(3)
2 s−1
The unit of frequency is the Hertz (Hz) or s−1 .
Exercise 3: Period and Frequency
What is the period of a wave of frequency 10 Hz?
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(Solution on p. 10.)
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2.6 Speed of a Transverse Wave
In Motion in One Dimension, we saw that speed was dened as
distance travelled
.
(4)
time taken
The distance between two successive peaks is 1 wavelength, λ. Thus in a time of 1 period, the wave will
travel 1 wavelength in distance. Thus the speed of the wave, v , is:
speed =
distance travelled λ
= .
time taken
T
1
However, f = T . Therefore, we can also write:
(5)
v=
v
=
λ
T
= λ·
=
(6)
1
T
λ·f
We call this equation the wave equation. To summarise, we have that v = λ · f where
• v = speed in m · s−1
• λ = wavelength in m
• f = frequency in Hz
Exercise 4: Speed of a Transverse Wave 1
(Solution on p. 10.)
Exercise 5: Speed of a Transverse Wave 2
(Solution on p. 11.)
When a particular string is vibrated at a frequency of 10 Hz, a transverse wave of wavelength
0, 25 m is produced. Determine the speed of the wave as it travels along the string.
A cork on the surface of a swimming pool bobs up and down once every second on some ripples.
The ripples have a wavelength of 20 cm. If the cork is 2 m from the edge of the pool, how long does
it take a ripple passing the cork to reach the edge?
The following video provides a summary of the concepts covered so far.
Khan academy video on waves - 1
This media object is a Flash object. Please view or download it at
<http://www.youtube.com/v/tJW_a6JeXD8&rel=0>
Figure 12
2.6.1 Waves
1. When the particles of a medium move perpendicular to the direction of the wave motion, the wave is
called a ......... wave.
Click here for the solution.1
1 http://www.fhsst.org.za/liq
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2. A transverse wave is moving downwards. In what direction do the particles in the medium move?
Click here for the solution.2
3. Consider the diagram below and answer the questions that follow:
Figure 13
a. the wavelength of the wave is shown by letter
b. the amplitude of the wave is shown by letter
.
.
Click here for the solution.3
4. Draw 2 wavelengths of the following transverse waves on the same graph paper. Label all important
values.
a. Wave 1: Amplitude = 1 cm, wavelength = 3 cm
b. Wave 2: Peak to trough distance (vertical) = 3 cm, peak to peak distance (horizontal) = 5 cm
Click here for the solution.4
5. You are given the transverse wave below.
Figure 14
Draw the following:
a. A wave with twice the amplitude of the given wave.
b. A wave with half the amplitude of the given wave.
c. A wave travelling at the same speed with twice the frequency of the given wave.
d. A wave travelling at the same speed with half the frequency of the given wave.
e. A wave with twice the wavelength of the given wave.
f. A wave with half the wavelength of the given wave.
g. A wave travelling at the same speed with twice the period of the given wave.
h. A wave travelling at the same speed with half the period of the given wave.
Click here for the solution.5
6. A transverse wave travelling at the same speed with an amplitude of 5 cm has a frequency of 15 Hz.
The horizontal distance from a crest to the nearest trough is measured to be 2,5 cm. Find the
a. period of the wave.
b. speed of the wave.
Click here for the solution.6
2 http://www.fhsst.org.za/li4
3 http://www.fhsst.org.za/li2
4 http://www.fhsst.org.za/lr8
5 http://www.fhsst.org.za/lr9
6 http://www.fhsst.org.za/lrX
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7. A y aps its wings back and forth 200 times each second. Calculate the period of a wing ap.
Click here for the solution.7
8. As the period of a wave increases, the frequency increases/decreases/does not change.
Click here for the solution.8
9. Calculate the frequency of rotation of the second hand on a clock.
Click here for the solution.9
10. Microwave ovens produce radiation with a frequency of 2 450 MHz (1 MHz = 106 Hz) and a wavelength
of 0,122 m. What is the wave speed of the radiation?
Click here for the solution.10
11. Study the following diagram and answer the questions:
Figure 15
a. Identify two sets of points that are in phase.
b. Identify two sets of points that are out of phase.
c. Identify any two points that would indicate a wavelength.
Click here for the solution.11
12. Tom is shing from a pier and notices that four wave crests pass by in 8 s and estimates the distance
between two successive crests is 4 m. The timing starts with the rst crest and ends with the fourth.
Calculate the speed of the wave.
Click here for the solution.12
7 http://www.fhsst.org.za/lrl
8 http://www.fhsst.org.za/lr5
9 http://www.fhsst.org.za/lrN
10 http://www.fhsst.org.za/lrR
11 http://www.fhsst.org.za/lrn
12 http://www.fhsst.org.za/lrQ
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Solutions to Exercises in this Module
Solution to Exercise (p. 4)
Step 1. The denition of the amplitude is the height of a peak above the equilibrium position. The still water
mark is the height of the water at equilibrium and the peak is 2 m above this, so the amplitude is 2 m.
Solution to Exercise (p. 5)
Step 1.
Figure 16
Step 2. From the sketch we see that 4 consecutive peaks is equivalent to 3 wavelengths.
Step 3. Therefore, the wavelength of the wave is:
3λ
=
6m
λ
=
6m
3
=
2m
(7)
Solution to Exercise (p. 6)
Step 1. We are required to calculate the period of a 10 Hz wave.
Step 2. We know that:
T =
1
f
(8)
Step 3.
T
=
1
f
1
10 Hz
=
0, 1 s
=
(9)
Step 4. The period of a 10 Hz wave is 0, 1 s.
Solution to Exercise (p. 7)
Step 1.
• frequency of wave: f = 10Hz
• wavelength of wave: λ = 0, 25m
We are required to calculate the speed of the wave as it travels along the string. All quantities are in
SI units.
Step 2. We know that the speed of a wave is:
(10)
v =f ·λ
Step 3.
and we are given all the necessary quantities.
v
=
f ·λ
=
(10 Hz) (0, 25 m)
=
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−1
2, 5 m · s
(11)
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Step 4. The wave travels at 2, 5 m · s−1 along the string.
Solution to Exercise (p. 7)
Step 1. We are given:
• frequency of wave: f = 1 Hz
• wavelength of wave: λ = 20 cm
• distance of cork from edge of pool: d = 2 m
We are required to determine the time it takes for a ripple to travel between the cork and the edge of
the pool.
The wavelength is not in SI units and should be converted.
Step 2. The time taken for the ripple to reach the edge of the pool is obtained from:
d
from v =
t
d
t=
v
(12)
We know that
v =f ·λ
(13)
d
f ·λ
(14)
Therefore,
t=
Step 3.
20 cm = 0, 2 m
(15)
Step 4.
=
d
f ·λ
2m
(1 Hz)(0,2 m)
=
10 s
t =
Step 5. A ripple passing the leaf will take 10 s to reach the edge of the pool.
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(16)